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When in high school I used to see mathematical objects as ideal objects whose existence is independent of us. But when I learned set theory, I discovered that all mathematical objects I was studying were sets, for example:

$ 0 = \emptyset $

What does it mean to say that the number $ 1 $ is the singleton set of the empty set?

Thank you all for the answers, it is helping me a lot.

Obs.

This question received far more attention than I expected it to do. After reading all the answers and reflecting on it for a while I've come to the following conclusions (and I would appreciate if you could add something to it or correct me): Let's take a familiar example, the ordered pair. We have an intuitive, naive notion of this 'concept' and of its fundamental properties like, for example, it has two components and

$ (x, y) = (a, b) $ iff $ x = a $ and $ y = b $

But mathematicians find it more convenient (and I agree) to define this concept in terms of set theory by saying that $ (x, y) $ is a shortcut for the set { {x}, {x, y} } and then proving the properties of the ordered pair. I.e. showing that this set-theoretical ordered pair has all the properties one expect the 'ideal' ordered pair to have. Secondly, mathematicians don't really care much about these issues.

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    $\begingroup$ It means that in set theory we can define (build-up) a "structure" (made of sets) that works as a model for the usual natural numbers, i.e. the said structure of sets has all the known properties of natural numbers. $\endgroup$ – Mauro ALLEGRANZA May 2 '18 at 13:52
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    $\begingroup$ Is the universe just a bunch of elementary particles? There is structure to the universe, and there is structure to mathematics, and studying that structure is what's interesting. $\endgroup$ – Lee Mosher May 2 '18 at 13:57
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    $\begingroup$ @LeeMosher I think the sense in which the universe is just a bunch of elementary particles is much stronger than the sense in which mathematical objects are built from the emptyset. $\endgroup$ – Noah Schweber May 2 '18 at 14:03
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    $\begingroup$ Put it this way: if a discrepancy were found between the set-theoretic formulation of arithmetic, and the way people have naively done arithmetic since prehistory, we would change the former, not the latter. $\endgroup$ – user360874 May 2 '18 at 18:17
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    $\begingroup$ Relevant and related quote from T. Tao: "At a purely formal level, one could call probability theory the study of measure spaces with total measure one, but that would be like calling number theory the study of strings of digits which terminate". $\endgroup$ – Aloizio Macedo May 3 '18 at 1:04

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Warning: personal opinion ahead!

It depends upon what the meaning of the word "is" is.

One way to think about it - not necessarily historically correct - is the following: that the axioms of set theory are intricate enough - or, if you prefer, describe structures (their models) which are intricate enough - to "implement" all of mathematics. This is analogous to the relation between algorithms (as clear-but-informal descriptions of processes) and programs (their actual implementation), with the observation that the same algorithm can be implemented in different ways.

It's worth noting that this "in different ways" has multiple senses: there will be many ways to write a program which carries out a specific algorithm in a given programming language, and there are also lots of programming languages. Correspondingly, we have:

  • There are lots of ways to implement (say) basic arithmetic inside ZFC; the von Neumann approach is just the standard one.

  • There are also different theories which similarly are intricate enough to implement all of mathematics.

Asking what $1$ "is" is an ontological question, but set theory doesn't need to be thought of ontologically - the pragmatic approach ("how can we formally and precisely implement mathematics?") is sufficient.


I'm not claiming that this is the universal view; for example, one can also argue (although I don't) that the cumulative hierarchy (= the sets "built from" the emptyset) consists of all the mathematical objects which are guaranteed to exist, in a Platonic sense. But I think the view above probably more faithfully reflects the general attitude of the mathematical community, and is certainly how I approach the question.

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    $\begingroup$ Great answer. I also think of ZFC and other formalisations of set theory as languages, which we use to make sentences. The sentences we make may not be perfectly the same as the thoughts we wish to express, but they come close enough for all practical purposes. Ultimately, language is not the same as thought, but we rely on it to communicate thoughts. $\endgroup$ – user159517 May 4 '18 at 9:35
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    $\begingroup$ Right; though historically OP's question goes back to before ZFC. Dedekind, Frege, Russell, Whitehead, and others wanted to reduce mathematics to "logic" via set theory (see en.wikipedia.org/wiki/Logicism). Russell's eponymous paradox largely killed the logicist project, then ZFC was invented to resolve some of the issues. Though it's harder to argue that ZFC is merely "part of logic", than that naive set theory is. Gödel also comes into it, but as Fermat said, Hanc marginis exiguitas non caperet. $\endgroup$ – TextGeek May 4 '18 at 15:51
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    $\begingroup$ "It depends upon what the meaning of the word "is" is" - are you certain you're not Bill Clinton?!? $\endgroup$ – Bob Jarvis May 6 '18 at 3:55
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Firstly, there is a big difference between abstract concepts and their concrete representations. It is true that we can encode and reason about natural numbers in ZFC, in the specific sense that we can reason about a model of PA (Peano Arithmetic) within ZFC. It does not mean that somehow the abstract concept of $0$ is the empty-set!

Similarly, it is true that we can express notions like "pair" and "function" and so on in terms of set-theoretic definitions in ZFC, but these notions have been around for far far longer than ZFC, and furthermore there are infinitely many viable definitions to 'make concrete' these notions in ZFC. No mathematician actually thinks of the abstract pair $\langle x , y \rangle$ as $\{\{x\},\{x,y\}\}$ or some other concrete representation.

Furthermore, even our standard mathematical notation shows that we conceive of functions in a more fundamental way than is suggested by the standard encoding as a set of input/output pairs.

For more details, see this post about abstract mathematical objects that are not sets, which also mentions urelements (non-sets such as you and me).


Secondly, it is true that from the viewpoint of ZFC, everything is a set (in the sense that given any two objects $x,y$ we can ask whether $x \in y$ or not). But even then, it is not necessarily the case that everything is "just a bunch of nested empty sets"!

The axiom of infinity is the only axiom of ZFC that asserts the (absolute) existence of some set. Every other axiom can be applied only if you already have some set. Now, informally the axiom of infinity says that there exists an inductive set, where a set $S$ is called inductive iff ( $S$ includes the empty set as a member, and is closed under the successor operation ), where the successor of $x$ is $S(x) := x \cup \{x\}$. Note that the axiom does not stipulate that there is a 'minimal' such set, nor what are the members of such a minimal set!

Well, we can use the other axioms of ZFC to construct $N$ to be the intersection of all inductive sets. But there is no way to prove that $N$ only includes as members the empty set and sets obtained by iteratively applying the successor operation. If that seems strange to you, that is unfortunately the way it is.

You see, ZFC does not have the natural numbers as primitive notions, and the axiom of infinity was concocted precisely to enable ZFC to construct a model of PA; we can define addition and multiplication on $N$ and prove that $N$ satisfies PA. But that means that $N$ is what we take to be natural numbers when working in ZFC! Nothing precludes 'our' set-theoretic universe (if it at all exists) from having an $N$ that has more members than those that you can manually write down, namely $0, S(0), S(S(0)), \cdots$, where $0 := \varnothing$.

And the curious thing about this is that ZFC itself knows that the above may happen! ZFC proves that if ZFC is consistent then there is a model of ZFC that has extra (called non-standard) members of $N$.


Finally, most mathematics is in fact independent of set theory, and can be recovered in very weak theories of arithmetic such as ACA, as briefly described here. For real-world applications, it is even better, because there is no evidence of infinitary objects in reality, and there is even a humorous grand conjecture by Harvey Friedman:

Every theorem published in the Annals of Mathematics whose statement involves only finitary mathematical objects (i.e., what logicians call an arithmetical statement) can be proved in EFA. (EFA is a fragment of PA.)

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In addition to Noah's answer: Viewing mathematics as a "bunch of nested empty sets" is insofar simplifying as mathematics is more than the objects used in mathematics. Logical deduction is not a nested empty set (although logical models can be created in this manner). On the other hand, given a strong enough axiomatic system to start with, all mathematical objects (describable in that system) can be modeled with nested empty sets. This isn't just a theory which is ignored in practice, but there are in fact proof assistants with their library built on that to a certain degree.

For example, in Mizar, starting from Tarski–Grothendieck set theory (which implies ZFC), the empty set is formally defined, $0$ is defined to equal the empty set and the natural numbers are built from scratch and fit the von Neumann approach. But it doesn't stop: the integers, rationals, real and complex numbers are all built up from there, in a slightly sophisticated way to ensure that the usual inclusions $\mathbb{N}\subset\mathbb{Z}\subset\mathbb{Q}\subset\mathbb{R}\subset\mathbb{C}$ hold. In this particular description, $-\frac{1}{2}i$ does unfold to $$ (0,1)\mapsto(0,-\tfrac{1}{2})=\{(0,0),(1,-\tfrac{1}{2})\} = \{\{\{0\}\},\{\{1,-\tfrac{1}{2}\},\{1\}\}\}$$ (with $(a,b)=\{\{a,b\},\{a\}\}$ and so $(a,a)=\{\{a\}\}$) and even further to $$ \{\{\{0\}\},\{\{\{0\},\{\{0,\{\{\{0\},\{0,\{0\}\}\},\{\{0\}\}\}\},\{0\}\}\},\{\{0\}\}\}\}$$ (using 9 zeros/empty sets) or, with $0$ as $\{\}$: $$ \{\{\{\{\}\}\},\{\{\{\{\}\},\{\{\{\},\{\{\{\{\}\},\{\{\},\{\{\}\}\}\},\{\{\{\}\}\}\}\},\{\{\}\}\}\},\{\{\{\}\}\}\}\}$$

But mathematicians tend to shorten the notation, so we stick with $-\frac{1}{2}i$ or $\frac{-i}{2}$ (which would be -1/2*<i> (or -2"*<i>) or -<i>/2 in Mizar).


On another note, in his German textbook "Einführung in die Mengenlehre" ("Introduction to set theory") 2nd ed. Oliver Deiser writes on p.42f

Die Mengenlehre ist hinsichtlich der Interpretation der gesamten Mathematik konkurrenzlos. Entscheidend ist hier nicht ein platonischer Glaube an die Mengen, sondern die Leistungsfähigkeit der Theorie und die Universalität der verwendeten Sprache.

which roughly translates to

Set theory is unrivaled regarding the interpretation of all of mathematics. The vital aspect here is not a platonic belief in sets but the power of the theory and the universality of the used language.

He writes this while having a critical look at the just naively introduced basics of set theory. He does so by dividing the critical look into three parts:

  1. "Was ist eigentlich ein mathematisches Objekt?" ("What is a mathematical object, really?")
  2. "Welche mathematischen Objekte (=Mengen) existieren?" ("Which mathematical objects (=sets) do exist?")
  3. "Was genau ist ein mathematischer Beweis einer Aussage?" ("What exactly is a mathematical proof of a statement?")

Deiser's book additionally has a lot of historical notes on the development on set theory, so it may be worth a look if you can get it in your language.

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    $\begingroup$ In CS, they call it an "abstraction leak" when features of an implementation are exposed. For instance, while every mathematical object can be encoded as nested empty sets, it ought to be a type/category error to speak about the elements of $-\frac{1}{2}i$, at least outside of actually implementing the complex numbers. $\endgroup$ – Kyle Miller May 2 '18 at 20:11
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    $\begingroup$ @KyleMiller True. But in math this "abstraction leak" is merely proof of its axiomatisation and rigorous formalization. In practice, besides initial technical work, the (non real) complex numbers in Mizar are not treated as Functions from $\{0,1\}$ to $\mathbb{R}$ but simply as complex numbers. So far I can't see how this "abstraction leak" could be evaluated as bad in this context, i.e. I see no potential for abuse. $\endgroup$ – SK19 May 2 '18 at 20:28
  • $\begingroup$ Is there any value to this nested set representation? Are there things you can do with nested sets that you can't do with other, more terse, numeric representations? $\endgroup$ – HackSlash May 3 '18 at 17:40
  • $\begingroup$ @HackSlash None that I know of. In programming, using a more basic representation can, under certain circumstances, speed up the program. But there is no intrinsic proof time associated with mathematical theorems. If anything, a proof checker may take longer to recognize the extended pattern (just as humans), which is not what we want at all. One could start a theory about how many empty sets are needed to represent a specific object, but apart from that I don't see any value besides a certain reassurance of everything working out fine (something mathematicians struggled with for a while) $\endgroup$ – SK19 May 3 '18 at 19:07
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    $\begingroup$ @HackSlash In programming terms, it's like compiling C code to Brainfuck code. It has zero practical value other than: 1) it's a neat trick, 2) it demonstrates that everything which can be done in C can be done in Brainfuck (which is useful if you're studying the computational power of Brainfuck) and 3) in the extremely unlikely event that you're somehow stuck in a world where you can only use Brainfuck, now you can also use C. $\endgroup$ – immibis May 4 '18 at 5:11
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Since this question is tagged with 'philosophy' I am surprised no one has mentioned Benacerraf's identification problem (you can read a summary here).

In short, there are infinitely many ways to identify the natural numbers with sets. Different ways of doing this lead to different and incompatible theorems in the meta-theory you would like to use to talk about the theory in which you defined natural numbers.

Philosophically, the conclusion that can be drawn is that, contrary to the statement in your question, all mathematical objects are not sets (since not even natural numbers are sets, except in a very specific sense of the word 'are').

There are of course many subtleties related to this issue, and the nature of what exactly mathematical objects are is a perennial one. Practicing mathematicians often seem to prefer Platonism.

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It means that people have been trying to boil everything down to the least possible amount of data, rules, and symbols. Axiomatisation is a part of this process.

But besides this, unless you really are studying the foundations for some purpose, it shouldn't really bother you or change the way you think of more advanced mathematics. No one is ever going to rewrite functional analysis textbooks from a point of view that acknowledges the fact that at the very beginning, $0$ is $\emptyset$.

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    $\begingroup$ It's not a fact. At the very beginning, $0$ is not $\emptyset$. It's $0$. $\endgroup$ – tomasz May 2 '18 at 21:49
  • $\begingroup$ @tomasz Yes it is, at least in the set-theoretic definition of numbers that the OP is referring to. $\endgroup$ – Arnaud Mortier May 2 '18 at 22:25
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    $\begingroup$ @ArnaudMortier But that's only "at the beginning" if one takes ZFC (and the specific von Neumann implementation of the naturals) as fundamental, which one needn't do. $\endgroup$ – Noah Schweber May 2 '18 at 22:31
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    $\begingroup$ @Noah Absolutely. But I understood that this is the settings of the OP. $\endgroup$ – Arnaud Mortier May 3 '18 at 1:22
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Your question is very related to the question of tautologies in mathematics. If from the set of axioms everything mathematicians develop are equivalent statements and implications, then all mathematics seem futile, as the axioms already contain all the truth you may ever find in them.

This of course is not a reasonable view of mathematics: just like physicists develop very different views of the universe according to the scales (standard model/quantum physics/nanostructures/chemistry/fluid mechanics/gravitation...) and are interested in the expressions and interactions of these laws in different contexts, mathematicians elaborate powerful abstractions that efficiently capture all necessary properties of mathematics to address a particular problem.

That natural numbers can be constructed from reasonable forms of set theory is reassuring; but this construction by no means pretends to be a suitable abstraction for most applications of mathematics.

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Only a logician thinks $0 = \varnothing$, $1 = \{0\}$, $2 = \{0,1\}$ and so on. "Genuine" mathematicians think of the "genuine" natural numbers and not of that baby model of them.

In fact, even for logicians, there are alternate models. Surreal numbers, say. So \begin{align*} 0 &= \{\;\;\;|\;\;\}, \\ 1 &= \{0\;|\;\;\;\}, \\ 2 &= \{1\;|\;\;\;\} \\ -1 &= \{\;\;\;|\;0\} \end{align*} and so on.

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    $\begingroup$ I didn't know we weren't genuine mathematicians. :P $\endgroup$ – Noah Schweber May 2 '18 at 14:26
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    $\begingroup$ Yeah get out of here poser @NoahSchweber $\endgroup$ – galois May 2 '18 at 15:16
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    $\begingroup$ I must be neither a logician nor a genuine mathematician, because I don't know how to tell if an instance of the natural numbers is genuine or not. Do I bite into it, like a coin, or what? $\endgroup$ – Rahul May 2 '18 at 17:42
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    $\begingroup$ Logician here: I definitely don’t believe $0 = \emptyset$, etc. (or any similar alternative implementation). $\endgroup$ – Peter LeFanu Lumsdaine May 3 '18 at 7:51
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    $\begingroup$ -1. This is not a genuine answer. $\endgroup$ – Mikhail Katz May 3 '18 at 9:15
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No, mathematics is not a bunch of nested empty sets. The OP is referring to a formalisation known as ZFC (Zermelo-Fraenkel set theory with the axiom of choice). This is an extremely valuable theory. However, only mathematicians think that $0 = \varnothing$, $1 = \{0\}$, $2 = \{0,1\}$ and so on. They rely on people like von Neumann to have done the foundational work for them, and assume that everything has been taken care of in ZFC, period. Genuine logicians know that ZFC is a far more powerful system than is needed to account for natural numbers, and usually work with a variety of weaker systems of which perhaps the most famous is Peano Arithmetic. The trouble with ZFC is that it requires one to take a lot more "on faith" than is necessary. An attitude common among mathematicians (and some set theorists as well) is one of foundationalism, i.e., believing that ZFC axioms are "true" and also believing in the existence of a Santa Cl... sorry, an intended interpretation.

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  • $\begingroup$ I don't assume but believe :) $\endgroup$ – SK19 May 5 '18 at 12:00
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To return to the original question of why one should consider 0 = Ø and 1 = {Ø}, there is a pragmatic answer. From ancient times up to the nineteenth century, mathematical objects were considered to be abstractions of objects out there: natural numbers, addition, multiplication, points, lines, vectors, etc. This gave rise to "silly" disputes on the existence of negative numbers, imaginary numbers, or infinite sets: they seem not to be out there.

To serve "physical" sciences, mathematics should not depend on physical reality. So mathematical objects were reduced to synthetic objects (symbols, actually) in the most economical way, and their basic properties were secured in axioms. Once you get convinced that it works, you can open up your natural intuition and do business as usual.

Leave the technical aspects to the professional mathematicians. Note that computer programs are quite happy with synthetic objects: they love 0's and 1's and anything you can build with it.

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This question received far more attention than i expected it to do. After reading all the answers and reflecting about it for a while i've come to the following conclusions (and i would apreciate if you could add something to it or correct me): Let's take a familiar example, the ordered pair. We have a intuitive, naive notion of this 'concept' and of it's fundamental properties like, for example it has two components and

$ (x, y) = (a, b) $ iff $ x = a $ and $ y = b $

But mathematicians find it more convenient (and i agree) to define this concept in terms of set theory by saying that

$ (x, y) $ is a shortcut for the set { {x}, {x, y} }

And then proving the properties of the ordered pair.

And secondly that mathematicians doesn't really care much for these issues.

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  • $\begingroup$ I'd recommend making this an addendum in an edit to your question, rather than an answer. $\endgroup$ – Christian Sykes May 7 '18 at 18:13

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